# Graphs of edge-to-vertex detour number 2

## Authors

• A. P. Santhakumaran Hindustan Institute of Technology and Science.

## Keywords:

Detour, Edge-to-vertex detour set, Edge-to-vertex detour basis, Edge-to- vertex detour number

## Abstract

For two vertices u and v in a graph G = (V,E), the detour distance D(u, v) is the length of a longest u − v path in G. A u − v path of length D(u, v) is called a u − v detour. For subsets A and B of V , the detour distance D(A,B) is defined as D(A,B) = min{D(x, y) : x ∈ A, y ∈ B}. A u − v path of length D(A,B) is called an A-B detour joining the sets A,B ⊆ V where u ∈ A and v ∈ B. A vertex x is said to lie on an A − B detour if x is a vertex of some A−B detour. A set S ⊆ E is called an edge-to-vertex detour set if every vertex of G is incident with an edge of S or lies on a detour joining a pair of edges of S. The edge-to-vertex detour number dn2(G) of G is the minimum order of its edge-to-vertex detour sets and any edge-to-vertex detour set of order dn2(G) is an edge-to-vertex detour basis of G. Graphs G of size q for which dn2(G) = 2 are characterized.

## Author Biography

### A. P. Santhakumaran, Hindustan Institute of Technology and Science.

Dept. of Mathematics.

## References

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A. P. Santhakumaran, “On the edge-to-vertex detour number of a graph”, Advanced studies in contemporary mathematics, vol. 24, no. 3, pp. 405-420, 2014.

## Published

2021-06-16 — Updated on 2021-07-26

## How to Cite


A. P. Santhakumaran, “Graphs of edge-to-vertex detour number 2”, Proyecciones (Antofagasta, On line), vol. 40, no. 4, pp. 963-979, Jul. 2021.

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